## 经济代写|宏观经济学代写Macroeconomics代考|The future of growth

The forces highlighted in these models of innovation, and their policy implications, have huge consequences for what we think will happen in the future when it comes to economic growth. There is a case for optimism, but also for its opposite.

Consider the first. If scale effects are present on a global scale, then as the world gets bigger growth will be faster, not the other way around. To see this, it is worth looking at the Kremer (1993) model in more detail, in a slightly simplified version. Consider the production function in
$$Y=A p^\alpha T^{1-\alpha}=A p^\alpha,$$
where $p$ is population and $T$ is land which is available in fixed supply which, for simplicity, we will assume is equal to 1 . We can rewrite it as
$$y=A p^{\alpha-1} .$$
The population dynamics have a Malthusian feature in the sense that they revert to a steady state that is sustainable given the technology. In other words, population adjusts to technology so that output per capita remains at subsistence level; as in the Malthusian framework, all productivity gains translate into a larger population, not into higher standards of living. (This is usually thought of as a good description of the pre-industrial era, as we will discuss in detail in Chapter 10.)
$$p=\left(\frac{\bar{y}}{A}\right)^{\frac{1}{x-1}} .$$
Critically, the scale effects come into the picture via the assumption that
$$\frac{\dot{A}}{A}=p g .$$
i.e. the rate of technological progress is a function of world population, along the lines of the endogenous growth models we have seen. We can now solve for the dynamics of population, using (6.23) and then (6.24):
$$\begin{gathered} \ln p=\left(\frac{1}{\alpha-1}\right)[\ln \bar{y}-\ln A], \ \frac{\dot{p}}{p}=-\left(\frac{1}{\alpha-1}\right) \frac{\dot{A}}{A}=\frac{1}{1-\alpha} \frac{\dot{A}}{A}=\frac{1}{1-\alpha} p g, \ \frac{\dot{p}}{p}=\left(\frac{g}{1-\alpha}\right) p . \end{gathered}$$

## 经济代写|宏观经济学代写Macroeconomics代考|Growth accounting

This is another founding contribution of Robert Solow to the study of economic growth. Right after publishing his “Contribution to the Theory of Economic Growth” in 1956, he published another article in 1957 (Solow 1957) noting that an aggregate production function such as
$$Y(t)=A(t) F\left(K_t, L_t\right),$$
when combined with competitive factor markets, immediately yields a framework that lets us account for the (proximate) sources of economic growth. Take the derivative of the log of the production function with respect to time,
$$\begin{gathered} \frac{\dot{Y}}{Y}=\frac{\dot{A}}{A}+\frac{A F_K}{Y} K+\frac{A F_L}{Y} L \Rightarrow \ \frac{\dot{Y}}{Y}=\frac{\dot{A}}{A}+\frac{A F_K K}{Y} \frac{\dot{K}}{K}+\frac{A F_L L}{Y} \frac{\dot{L}}{L} \Rightarrow \ g_Y=g_A+\alpha_K g_K+\alpha_L g_L, \end{gathered}$$
where $g_X$ is the growth rate of variable $X$, and $\alpha_X \equiv \frac{A F_X X}{X}$ is the elasticity of output with respect to factor $X$. This is an identity, but adding the assumption of competitive factor markets (i.e. factors are paid their marginal productivity) means that $\alpha_X$ is also the share of output that factor $X$ obtains as payment for its services. Equation (7.2) then enables us to estimate the contributions of factor accumulation and technological progress (often referred to as total factor productivity (TFP)) to economic growth.
This is how it works in practice: from national accounts and other data sources, one can estimate the values of $g_Y, g_K, g_L, \alpha_K$, and $\alpha_L$; from (7.2) one can then back out the estimate for $g_{A^{-}}{ }^2$ (For this reason, $g_A$ is widely referred to as the Solow residual.) Solow actually computed this for the U.S. economy, and reached the conclusion that the bulk of economic growth, about $2 / 3$, could be attributed to the residual. Technological progress, and not factor accumulation, seems to be the key to economic growth.

Now, here is where a caveat is needed: $g_A$ is calculated as a residual, not directly from measures of technological progress. It is the measure of our ignorance! ${ }^3$ More precisely, any underestimate of the increase in $K$ or $L$ (say, because it is hard to adjust for the increased quality of labour input), will result in an overestimate of $g_A$. As a result, a lot of effort has been devoted to better measure the contribution of the different factors of production.

## 经济代写|宏观经济学代写Macroeconomics代考|The future of growth

$$Y=A p^\alpha T^{1-\alpha}=A p^\alpha,$$

$$y=A p^{\alpha-1} .$$

$$p=\left(\frac{\bar{y}}{A}\right)^{\frac{1}{x-1}} .$$

$$\frac{\dot{A}}{A}=p g .$$

$$\ln p=\left(\frac{1}{\alpha-1}\right)[\ln \bar{y}-\ln A], \frac{\dot{p}}{p}=-\left(\frac{1}{\alpha-1}\right) \frac{\dot{A}}{A}=\frac{1}{1-\alpha} \frac{\dot{A}}{A}=\frac{1}{1-\alpha} p g, \frac{\dot{p}}{p}=\left(\frac{g}{1-\alpha}\right) p .$$

## 经济代写|宏观经济学代写Macroeconomics代考|Growth accounting

$$Y(t)=A(t) F\left(K_t, L_t\right),$$

$$\frac{\dot{Y}}{Y}=\frac{\dot{A}}{A}+\frac{A F_K}{Y} K+\frac{A F_L}{Y} L \Rightarrow \frac{\dot{Y}}{Y}=\frac{\dot{A}}{A}+\frac{A F_K K}{Y} \frac{\dot{K}}{K}+\frac{A F_L L}{Y} \frac{\dot{L}}{L} \Rightarrow g_Y=g_A+\alpha_K g_K+\alpha_L g_L$$

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